First slide

Theory of equations

Question

If $α, β$ are the roots of $(x - a) (x - b) + c = 0$ ,
$c \neq 0$ ,then roots of $(α β - c) x^{2} + (α + β) x + 1 = 0$ are

Moderate

A

1/a, 1/b
B

– 1/a, – 1/b
C

1/a, –1/b
D

–1/a, 1/b

Solution

$α, β$ are roots of $x^{2} - (a + b) x + a b + c = 0$
$∴ α + β = a + b, α β = a b + c$
The equation
$(α β - c) x^{2} + (α + β) x + 1 = 0$
becomes
$\begin{array}{l} a b x^{2} + (a + b) x + 1 = 0 \\ \Rightarrow (a x + 1) (b x + 1) = 0 \\ \Rightarrow x = - 1 / a, - 1 / b \end{array}$

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